One of the greatest achievements of modern mathematics is the classification of finite simple groups, the culmination of over 10,000 pages of mathematical proof. Groups are the canonical algebraic objects for describing symmetries. A natural question is then “What are all the groups that exist?” which is to say “What are all the possible ways something can have symmetry?” This question turns out to be much too difficult to answer, even restricting our attention to groups consisting of finitely many symmetries. However, finite groups can be broken down into “finite simple groups”. Being analogous to atoms, their classification is analogous to a periodic table for group theory.
The natural way to conceptualise a group is to realise it as the symmetries of some canonical object. For the vast majority of finite simple groups this is easily done, with 18 countably infinite families enumerated as the symmetries of commensurate families of easily constructed objects. The simplest example are the cyclic groups of prime order, consisting of the rotational symmetries of a regular polygon with a prime number of sides.
What’s more astonishing are the remaining 26 groups which do not fall into any infinite family and do not follow any regular pattern, collectively known as the sporadic groups. The largest of these is the Monster, which contains about 10⁵⁶ symmetries. There is a sense in which the Monster contains 20 of the sporadic groups including itself which Robert Griess called the happy family, the remaining 6 being called the pariahs.
The sporadic groups, themselves exceptional objects, tend to arise as the symmetries of other exceptional objects. The earliest known sporadic groups, called the Mathieu groups, arise as the automorphisms of an exceptional error-correcting code: The 24-bit Golay code. By removing one bit of the 24-bit Golay code one obtains the 23-bit Golay code, whose astounding error-correcting capabilities lead to its use in the Voyager 1 and 2 spacecraft. The Golay code can itself be used to construct the Leech lattice, which represents the best known sphere packing in 24 dimensions. The symmetries of the Leech lattice in turn give rise to the Conway groups.
The Mathieu groups, the Conway groups, and the Monster form a tower of exceptional groups, each in a sense containing the last. What, then, should crown the corresponding tower of canonical exceptional objects upon which they act?
The existence of the Monster was originally proven by Griess by exhibiting it as the symmetries of a 196,884 dimensional object. The construction of the so-called Griess algebra was later simplified by Tits and Conway, but this still lacked the elegance of the Golay code and Leech lattice, both of which possess relatively simple characterisations.
Even before Griess’s construction confirmed the existence of the Monster, mathematicians had already begun collecting an astounding number of facts relating the group to seemingly disparate areas of mathematics — facts which came to be known as Monstrous Moonshine, a term due to Conway, and which would provide clues for the construction of a canonical space acted on by the Monster.
James Jensen
The University of Western Australia
